Constructing elliptic curves of prime order

TL;DR

This paper introduces an efficient algorithm for constructing elliptic curves with a specified prime order, significantly advancing the ability to generate such curves quickly for cryptographic applications.

Contribution

The paper presents a polynomial-time algorithm for constructing elliptic curves of prime order, incorporating modular functions and analyzing theoretical limits on optimization.

Findings

Algorithm runs in heuristic polynomial time Otilde((log N)^3)

Use of modular functions reduces runtime constants

Gonality bounds limit potential runtime improvements

Abstract

We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained.